Observability of Linear Hybrid Systems

1
Introduction
Observability refers to the study of the conditions under which it is possible to uniquely infer the state of a dynamical system from measurements of its output. For discrete-event systems, the definition of current-location observability was proposed in [12] as the property of being able to estimate the location of the system after a finite number of steps. A similar definition was given in [11] together with a polynomial test for observability, the so-called current-location tree, which depends on properties of the nodes of a finite state machine associated with the discrete-event system. For continuous systems with linear dynamics, it is well known that the observability problem can be reduced to that of analyzing the rank of the so-called observability matrix. This is the well known Popov-BelevicHautus rank test for linear systems [10]. For nonlinear systems with smooth dynamics, different definitions of observability have been proposed. We refer interested readers to [8] and references therein for a recent comparison of different definitions of observability. For hybrid systems, most of the previous work has concentrated on the areas of modeling, stability, controllability and verification (see previous workshop proceedings). Relatively little attention, however, has been devoted to the study of the observability of both the continuous and discrete states of a hybrid system.
Abstract. We analyze the observability of the continuous and discrete states of continuous-time linear hybrid systems. For the class of jumplinear systems, we derive necessary and sufficient conditions that the structural parameters of the model must satisfy in order for filtering and smoothing algorithms to operate correctly. Our conditions are simple rank tests that exploit the geometry of the observability subspaces. For linear hybrid systems, we derive weaker rank conditions that are sufficient to guarantee the uniqueness of the reconstruction of the state trajectory, even when the individual linear systems are unobservable the best of our knowledge, the first attempt to characterize the observability of hybrid systems can be found in [15], where a definition of observability is proposed. [14] gives conditions for the observability of a particular class of linear time-varying systems where the system matrix is a linear combination of a basis with respect to time-varying coefficients. [6] addresses the observability and controllability of switched linear systems with known and periodic transitions. [13] gives a condition for the observability of switched linear systems in terms of the existence of a discrete state trajectory. [3] proposes the notion of incremental observability for piecewise affine systems. Such a notion requires the solution of a mixed-integer linear program in order to be tested. [16] derives necessary and sufficient conditions for the observability of discrete-time jump-linear systems. The conditions can be tested using simple simple rank tests on the structural parameters of the model. [5] derives different rank tests for the weak observability of jump-Markov linear systems. [9] gives observability conditions for stochastic linear hybrid systems in terms of the covariances of the outputs. A problem related to observability that has been recently considered is the design of observers for linear hybrid systems. [1] considers the case in which the discrete state is known and proposes a Luenberger observer for the continuous state. [2] combines location observers with Luenberger observers to design a hybrid observer that identifies the discrete location in a finite number of steps and converges exponentially to the continuous state. 1.2 Contributions
In this paper we study the observability of a class of linear hybrid systems known as jump- (or switched-) linear systems, i.e., systems whose evolution is determined by a collection of linear models with continuous state x t ∈ Rn connected by switches among a number of discrete states λt ∈ {1, 2, . . . , N }. In Section 2 we introduce a notion of observability for jump-linear systems. We define the observability index ν of a jump-linear system and use it to derive rank conditions that the structural parameters of the model must satisfy in order for filtering and smoothing algorithms to operate correctly. We show that the state trajectory is observable if and only if the pairwise intersection of different observable subspaces is trivial. We also show that the switching times are observable if and only if the difference between any pair of observability matrices is nonsingular. The rank conditions we derive are simpler than their discrete-time counterparts [16] and can be thought of as an extension of the Popov-Belevic-Hautus rank test for linear systems. Our conditions only depend on the geometry of the observability subspaces, and therefore they are applicable also to the case of hybrid models where the switching mechanism depends on the continuous state. In Section 3 we derive weaker rank conditions that guarantee the observability of a linear hybrid system, even if the individual linear systems are unobservable. In this case, observability is gained by requiring that the output switches at least once in the given observability interval. Section 4 concludes the paper with a discussion about the role of the inputs in the observability of the discrete state.